Monotonicity Tests for Signal Deciles

The standard way to eyeball a signal is to sort stocks into ten deciles by signal value each month, then look at the average return of decile 10 minus decile 1. A big, positive gap looks like evidence the signal works. But that single comparison can hide a lot: what if the returns actually rise smoothly across deciles 1 through 7, then jump erratically, with decile 8 lower than decile 6? A large top-minus-bottom spread can be produced by a genuinely monotonic, well-behaved signal, or by a signal whose only real information is "the very top and very bottom deciles are different," with nothing systematic happening in between — two very different claims about how the signal works.

What monotonicity actually checks

A monotonicity test asks whether decile-average returns increase (or decrease) in the expected direction at every single step, not just between the extremes: is decile 2's average return higher than decile 1's, is decile 3's higher than decile 2's, and so on. This is often tested with a bootstrap-based test on the ordered sequence of decile means, checking whether a fully increasing sequence is unlikely to arise by chance, or with simpler nonparametric rank-based approaches.

A fully monotonic pattern says the signal captures something that scales gradually across its whole range — the behavior expected of a genuine, continuous relationship (cheaper stocks earning steadily higher returns as valuation decreases). A non-monotonic pattern, where the middle deciles are jumbled despite decile 10 beating decile 1, suggests the "signal" might just separate two extreme groups rather than measure a smooth underlying quantity — a materially less trustworthy story.

Worked example

Two signals both show decile 10 minus decile 1 average monthly returns of 1.2%. Signal A's ten decile averages, in order, are 0.1%, 0.2%, 0.35%, 0.5%, 0.6%, 0.75%, 0.9%, 1.0%, 1.1%, 1.3% — strictly increasing at every step. Signal B's are 0.1%, 0.3%, 0.9%, 0.4%, 0.6%, 0.5%, 0.8%, 0.7%, 1.0%, 1.3% — the same start and end point, and roughly the same average slope, but decile 3 exceeds decile 4, and decile 7 exceeds deciles 8 and 9 come back down before the final jump. Signal A passes a monotonicity check cleanly; Signal B would fail one, despite reporting an identical headline decile spread — a reader shown only the spread number would never know the two signals behave so differently in between.

What this means in practice

Monotonicity checks are a standard part of factor and signal due diligence precisely because the headline decile-spread statistic is easy to produce and easy to overstate: a wide spread built almost entirely from decile 10 or decile 1 alone, with everything in between flat or reversed, is a common and easily missed red flag in factor research. Reporting the full decile table (or its plot) alongside the top-minus-bottom spread, rather than the spread alone, is the simplest way to let a reader judge for themselves whether a signal behaves the way its headline number implies.